(************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) (* case_style -> (type_constraint -> env -> evar_map ref -> glob_constr -> unsafe_judgment) * evar_map ref -> type_constraint -> env -> glob_constr option * tomatch_tuples * cases_clauses -> unsafe_judgment end let rec list_try_compile f = function | [a] -> f a | [] -> anomaly "try_find_f" | h::t -> try f h with UserError _ | TypeError _ | PretypeError _ | PatternMatchingError _ | Loc.Exc_located (_, (UserError _ | TypeError _ | PretypeError _ | PatternMatchingError _)) -> list_try_compile f t let force_name = let nx = Name (id_of_string "x") in function Anonymous -> nx | na -> na (************************************************************************) (* Pattern-matching compilation (Cases) *) (************************************************************************) (************************************************************************) (* Configuration, errors and warnings *) open Pp let msg_may_need_inversion () = strbrk "Found a matching with no clauses on a term unknown to have an empty inductive type." (* Utils *) let make_anonymous_patvars n = list_make n (PatVar (dummy_loc,Anonymous)) (* Environment management *) let push_rels vars env = List.fold_right push_rel vars env (* We have x1:t1...xn:tn,xi':ti,y1..yk |- c and re-generalize over xi:ti to get x1:t1...xn:tn,xi':ti,y1..yk |- c[xi:=xi'] *) let relocate_rel n1 n2 k j = if j = n1+k then n2+k else j let rec relocate_index n1 n2 k t = match kind_of_term t with | Rel j when j = n1+k -> mkRel (n2+k) | Rel j when j < n1+k -> t | Rel j when j > n1+k -> t | _ -> map_constr_with_binders succ (relocate_index n1 n2) k t (**********************************************************************) (* Structures used in compiling pattern-matching *) type 'a rhs = { rhs_env : env; rhs_vars : identifier list; avoid_ids : identifier list; it : 'a option} type 'a equation = { patterns : cases_pattern list; rhs : 'a rhs; alias_stack : name list; eqn_loc : loc; used : bool ref } type 'a matrix = 'a equation list (* 1st argument of IsInd is the original ind before extracting the summary *) type tomatch_type = | IsInd of types * inductive_type * name list | NotInd of constr option * types type tomatch_status = | Pushed of ((constr * tomatch_type) * int list * name) | Alias of (name * constr * (constr * types)) | NonDepAlias | Abstract of int * rel_declaration type tomatch_stack = tomatch_status list (* We keep a constr for aliases and a cases_pattern for error message *) type pattern_history = | Top | MakeConstructor of constructor * pattern_continuation and pattern_continuation = | Continuation of int * cases_pattern list * pattern_history | Result of cases_pattern list let start_history n = Continuation (n, [], Top) let feed_history arg = function | Continuation (n, l, h) when n>=1 -> Continuation (n-1, arg :: l, h) | Continuation (n, _, _) -> anomaly ("Bad number of expected remaining patterns: "^(string_of_int n)) | Result _ -> anomaly "Exhausted pattern history" (* This is for non exhaustive error message *) let rec glob_pattern_of_partial_history args2 = function | Continuation (n, args1, h) -> let args3 = make_anonymous_patvars (n - (List.length args2)) in build_glob_pattern (List.rev_append args1 (args2@args3)) h | Result pl -> pl and build_glob_pattern args = function | Top -> args | MakeConstructor (pci, rh) -> glob_pattern_of_partial_history [PatCstr (dummy_loc, pci, args, Anonymous)] rh let complete_history = glob_pattern_of_partial_history [] (* This is to build glued pattern-matching history and alias bodies *) let rec pop_history_pattern = function | Continuation (0, l, Top) -> Result (List.rev l) | Continuation (0, l, MakeConstructor (pci, rh)) -> feed_history (PatCstr (dummy_loc,pci,List.rev l,Anonymous)) rh | _ -> anomaly "Constructor not yet filled with its arguments" let pop_history h = feed_history (PatVar (dummy_loc, Anonymous)) h (* Builds a continuation expecting [n] arguments and building [ci] applied to this [n] arguments *) let push_history_pattern n pci cont = Continuation (n, [], MakeConstructor (pci, cont)) (* A pattern-matching problem has the following form: env, evd |- match terms_to_tomatch return pred with mat end where terms_to_match is some sequence of "instructions" (t1 ... tp) and mat is some matrix (p11 ... p1n -> rhs1) ( ... ) (pm1 ... pmn -> rhsm) Terms to match: there are 3 kinds of instructions - "Pushed" terms to match are typed in [env]; these are usually just Rel(n) except for the initial terms given by user; in Pushed ((c,tm),deps,na), [c] is the reference to the term (which is a Rel or an initial term), [tm] is its type (telling whether we know if it is an inductive type or not), [deps] is the list of terms to abstract before matching on [c] (these are rels too) - "Abstract" instructions mean that an abstraction has to be inserted in the current branch to build (this means a pattern has been detected dependent in another one and a generalization is necessary to ensure well-typing) Abstract instructions extend the [env] in which the other instructions are typed - "Alias" instructions mean an alias has to be inserted (this alias is usually removed at the end, except when its type is not the same as the type of the matched term from which it comes - typically because the inductive types are "real" parameters) - "NonDepAlias" instructions mean the completion of a matching over a term to match as for Alias but without inserting this alias because there is no dependency in it Right-hand sides: They consist of a raw term to type in an environment specific to the clause they belong to: the names of declarations are those of the variables present in the patterns. Therefore, they come with their own [rhs_env] (actually it is the same as [env] except for the names of variables). *) type 'a pattern_matching_problem = { env : env; evdref : evar_map ref; pred : constr; tomatch : tomatch_stack; history : pattern_continuation; mat : 'a matrix; caseloc : loc; casestyle : case_style; typing_function: type_constraint -> env -> evar_map ref -> 'a option -> unsafe_judgment } (*--------------------------------------------------------------------------* * A few functions to infer the inductive type from the patterns instead of * * checking that the patterns correspond to the ind. type of the * * destructurated object. Allows type inference of examples like * * match n with O => true | _ => false end * * match x in I with C => true | _ => false end * *--------------------------------------------------------------------------*) (* Computing the inductive type from the matrix of patterns *) (* We use the "in I" clause to coerce the terms to match and otherwise use the constructor to know in which type is the matching problem Note that insertion of coercions inside nested patterns is done each time the matrix is expanded *) let rec find_row_ind = function [] -> None | PatVar _ :: l -> find_row_ind l | PatCstr(loc,c,_,_) :: _ -> Some (loc,c) let inductive_template evdref env tmloc ind = let arsign = get_full_arity_sign env ind in let hole_source = match tmloc with | Some loc -> fun i -> (loc, TomatchTypeParameter (ind,i)) | None -> fun _ -> (dummy_loc, InternalHole) in let (_,evarl,_) = List.fold_right (fun (na,b,ty) (subst,evarl,n) -> match b with | None -> let ty' = substl subst ty in let e = e_new_evar evdref env ~src:(hole_source n) ty' in (e::subst,e::evarl,n+1) | Some b -> (substl subst b::subst,evarl,n+1)) arsign ([],[],1) in applist (mkInd ind,List.rev evarl) let try_find_ind env sigma typ realnames = let (IndType(_,realargs) as ind) = find_rectype env sigma typ in let names = match realnames with | Some names -> names | None -> list_make (List.length realargs) Anonymous in IsInd (typ,ind,names) let inh_coerce_to_ind evdref env ty tyi = let expected_typ = inductive_template evdref env None tyi in (* devrait être indifférent d'exiger leq ou pas puisque pour un inductif cela doit être égal *) let _ = e_cumul env evdref expected_typ ty in () let binding_vars_of_inductive = function | NotInd _ -> [] | IsInd (_,IndType(_,realargs),_) -> List.filter isRel realargs let extract_inductive_data env sigma (_,b,t) = if b<>None then (NotInd (None,t),[]) else let tmtyp = try try_find_ind env sigma t None with Not_found -> NotInd (None,t) in let tmtypvars = binding_vars_of_inductive tmtyp in (tmtyp,tmtypvars) let unify_tomatch_with_patterns evdref env loc typ pats realnames = match find_row_ind pats with | None -> NotInd (None,typ) | Some (_,(ind,_)) -> inh_coerce_to_ind evdref env typ ind; try try_find_ind env !evdref typ realnames with Not_found -> NotInd (None,typ) let find_tomatch_tycon evdref env loc = function (* Try if some 'in I ...' is present and can be used as a constraint *) | Some (_,ind,_,realnal) -> mk_tycon (inductive_template evdref env loc ind),Some (List.rev realnal) | None -> empty_tycon,None let coerce_row typing_fun evdref env pats (tomatch,(_,indopt)) = let loc = Some (loc_of_glob_constr tomatch) in let tycon,realnames = find_tomatch_tycon evdref env loc indopt in let j = typing_fun tycon env evdref tomatch in let typ = nf_evar !evdref j.uj_type in let t = try try_find_ind env !evdref typ realnames with Not_found -> unify_tomatch_with_patterns evdref env loc typ pats realnames in (j.uj_val,t) let coerce_to_indtype typing_fun evdref env matx tomatchl = let pats = List.map (fun r -> r.patterns) matx in let matx' = match matrix_transpose pats with | [] -> List.map (fun _ -> []) tomatchl (* no patterns at all *) | m -> m in List.map2 (coerce_row typing_fun evdref env) matx' tomatchl (************************************************************************) (* Utils *) let mkExistential env ?(src=(dummy_loc,InternalHole)) evdref = e_new_evar evdref env ~src:src (new_Type ()) let evd_comb2 f evdref x y = let (evd',y) = f !evdref x y in evdref := evd'; y module Cases_F(Coercion : Coercion.S) : S = struct let adjust_tomatch_to_pattern pb ((current,typ),deps,dep) = (* Ideally, we could find a common inductive type to which both the term to match and the patterns coerce *) (* In practice, we coerce the term to match if it is not already an inductive type and it is not dependent; moreover, we use only the first pattern type and forget about the others *) let typ,names = match typ with IsInd(t,_,names) -> t,Some names | NotInd(_,t) -> t,None in let tmtyp = try try_find_ind pb.env !(pb.evdref) typ names with Not_found -> NotInd (None,typ) in match tmtyp with | NotInd (None,typ) -> let tm1 = List.map (fun eqn -> List.hd eqn.patterns) pb.mat in (match find_row_ind tm1 with | None -> (current,tmtyp) | Some (_,(ind,_)) -> let indt = inductive_template pb.evdref pb.env None ind in let current = if deps = [] & isEvar typ then (* Don't insert coercions if dependent; only solve evars *) let _ = e_cumul pb.env pb.evdref indt typ in current else (evd_comb2 (Coercion.inh_conv_coerce_to dummy_loc pb.env) pb.evdref (make_judge current typ) (mk_tycon_type indt)).uj_val in let sigma = !(pb.evdref) in (current,try_find_ind pb.env sigma indt names)) | _ -> (current,tmtyp) let type_of_tomatch = function | IsInd (t,_,_) -> t | NotInd (_,t) -> t let mkDeclTomatch na = function | IsInd (t,_,_) -> (na,None,t) | NotInd (c,t) -> (na,c,t) let map_tomatch_type f = function | IsInd (t,ind,names) -> IsInd (f t,map_inductive_type f ind,names) | NotInd (c,t) -> NotInd (Option.map f c, f t) let liftn_tomatch_type n depth = map_tomatch_type (liftn n depth) let lift_tomatch_type n = liftn_tomatch_type n 1 (**********************************************************************) (* Utilities on patterns *) let current_pattern eqn = match eqn.patterns with | pat::_ -> pat | [] -> anomaly "Empty list of patterns" let alias_of_pat = function | PatVar (_,name) -> name | PatCstr(_,_,_,name) -> name let remove_current_pattern eqn = match eqn.patterns with | pat::pats -> { eqn with patterns = pats; alias_stack = alias_of_pat pat :: eqn.alias_stack } | [] -> anomaly "Empty list of patterns" let push_current_pattern (cur,ty) eqn = match eqn.patterns with | pat::pats -> let rhs_env = push_rel (alias_of_pat pat,Some cur,ty) eqn.rhs.rhs_env in { eqn with rhs = { eqn.rhs with rhs_env = rhs_env }; patterns = pats } | [] -> anomaly "Empty list of patterns" let prepend_pattern tms eqn = {eqn with patterns = tms@eqn.patterns } (**********************************************************************) (* Well-formedness tests *) (* Partial check on patterns *) exception NotAdjustable let rec adjust_local_defs loc = function | (pat :: pats, (_,None,_) :: decls) -> pat :: adjust_local_defs loc (pats,decls) | (pats, (_,Some _,_) :: decls) -> PatVar (loc, Anonymous) :: adjust_local_defs loc (pats,decls) | [], [] -> [] | _ -> raise NotAdjustable let check_and_adjust_constructor env ind cstrs = function | PatVar _ as pat -> pat | PatCstr (loc,((_,i) as cstr),args,alias) as pat -> (* Check it is constructor of the right type *) let ind' = inductive_of_constructor cstr in if eq_ind ind' ind then (* Check the constructor has the right number of args *) let ci = cstrs.(i-1) in let nb_args_constr = ci.cs_nargs in if List.length args = nb_args_constr then pat else try let args' = adjust_local_defs loc (args, List.rev ci.cs_args) in PatCstr (loc, cstr, args', alias) with NotAdjustable -> error_wrong_numarg_constructor_loc loc (Global.env()) cstr nb_args_constr else (* Try to insert a coercion *) try Coercion.inh_pattern_coerce_to loc pat ind' ind with Not_found -> error_bad_constructor_loc loc cstr ind let check_all_variables typ mat = List.iter (fun eqn -> match current_pattern eqn with | PatVar (_,id) -> () | PatCstr (loc,cstr_sp,_,_) -> error_bad_pattern_loc loc cstr_sp typ) mat let check_unused_pattern env eqn = if not !(eqn.used) then raise_pattern_matching_error (eqn.eqn_loc, env, UnusedClause eqn.patterns) let set_used_pattern eqn = eqn.used := true let extract_rhs pb = match pb.mat with | [] -> errorlabstrm "build_leaf" (msg_may_need_inversion()) | eqn::_ -> set_used_pattern eqn; eqn.rhs (**********************************************************************) (* Functions to deal with matrix factorization *) let occur_in_rhs na rhs = match na with | Anonymous -> false | Name id -> List.mem id rhs.rhs_vars let is_dep_patt_in eqn = function | PatVar (_,name) -> occur_in_rhs name eqn.rhs | PatCstr _ -> true let mk_dep_patt_row (pats,_,eqn) = List.map (is_dep_patt_in eqn) pats let dependencies_in_pure_rhs nargs eqns = if eqns = [] then list_make nargs false (* Only "_" patts *) else let deps_rows = List.map mk_dep_patt_row eqns in let deps_columns = matrix_transpose deps_rows in List.map (List.exists ((=) true)) deps_columns let dependent_decl a = function | (na,None,t) -> dependent a t | (na,Some c,t) -> dependent a t || dependent a c let rec dep_in_tomatch n = function | (Pushed _ | Alias _ | NonDepAlias) :: l -> dep_in_tomatch n l | Abstract (_,d) :: l -> dependent_decl (mkRel n) d or dep_in_tomatch (n+1) l | [] -> false let dependencies_in_rhs nargs current tms eqns = match kind_of_term current with | Rel n when dep_in_tomatch n tms -> list_make nargs true | _ -> dependencies_in_pure_rhs nargs eqns (* Computing the matrix of dependencies *) (* [find_dependency_list tmi [d(i+1);...;dn]] computes in which declarations [d(i+1);...;dn] the term [tmi] is dependent in. [find_dependencies_signature (used1,...,usedn) ((tm1,d1),...,(tmn,dn))] returns [(deps1,...,depsn)] where [depsi] is a subset of n,..,i+1 denoting in which of the d(i+1)...dn, the term tmi is dependent. Dependencies are expressed by index, e.g. in dependency list [n-2;1], [1] points to [dn] and [n-2] to [d3] *) let rec find_dependency_list tmblock = function | [] -> [] | (used,tdeps,d)::rest -> let deps = find_dependency_list tmblock rest in if used && List.exists (fun x -> dependent_decl x d) tmblock then list_add_set (List.length rest + 1) (list_union deps tdeps) else deps let find_dependencies is_dep_or_cstr_in_rhs (tm,(_,tmtypleaves),d) nextlist = let deps = find_dependency_list (tm::tmtypleaves) nextlist in if is_dep_or_cstr_in_rhs || deps <> [] then ((true ,deps,d)::nextlist) else ((false,[] ,d)::nextlist) let find_dependencies_signature deps_in_rhs typs = let l = List.fold_right2 find_dependencies deps_in_rhs typs [] in List.map (fun (_,deps,_) -> deps) l (* Assume we had terms t1..tq to match in a context xp:Tp,...,x1:T1 |- and xn:Tn has just been regeneralized into x:Tn so that the terms to match are now to be considered in the context xp:Tp,...,x1:T1,x:Tn |-. [relocate_index_tomatch n 1 tomatch] updates t1..tq so that former references to xn1 are now references to x. Note that t1..tq are already adjusted to the context xp:Tp,...,x1:T1,x:Tn |-. [relocate_index_tomatch 1 n tomatch] will go the way back. *) let relocate_index_tomatch n1 n2 = let rec genrec depth = function | [] -> [] | Pushed ((c,tm),l,na) :: rest -> let c = relocate_index n1 n2 depth c in let tm = map_tomatch_type (relocate_index n1 n2 depth) tm in let l = List.map (relocate_rel n1 n2 depth) l in Pushed ((c,tm),l,na) :: genrec depth rest | Alias (na,c,d) :: rest -> (* [c] is out of relocation scope *) Alias (na,c,map_pair (relocate_index n1 n2 depth) d) :: genrec depth rest | NonDepAlias :: rest -> NonDepAlias :: genrec depth rest | Abstract (i,d) :: rest -> let i = relocate_rel n1 n2 depth i in Abstract (i,map_rel_declaration (relocate_index n1 n2 depth) d) :: genrec (depth+1) rest in genrec 0 (* [replace_tomatch n c tomatch] replaces [Rel n] by [c] in [tomatch] *) let rec replace_term n c k t = if isRel t && destRel t = n+k then lift k c else map_constr_with_binders succ (replace_term n c) k t let length_of_tomatch_type_sign na = function | NotInd _ -> if na<>Anonymous then 1 else 0 | IsInd (_,_,names) -> List.length names + if na<>Anonymous then 1 else 0 let replace_tomatch n c = let rec replrec depth = function | [] -> [] | Pushed ((b,tm),l,na) :: rest -> let b = replace_term n c depth b in let tm = map_tomatch_type (replace_term n c depth) tm in List.iter (fun i -> if i=n+depth then anomaly "replace_tomatch") l; Pushed ((b,tm),l,na) :: replrec depth rest | Alias (na,b,d) :: rest -> (* [b] is out of replacement scope *) Alias (na,b,map_pair (replace_term n c depth) d) :: replrec depth rest | NonDepAlias :: rest -> NonDepAlias :: replrec depth rest | Abstract (i,d) :: rest -> Abstract (i,map_rel_declaration (replace_term n c depth) d) :: replrec (depth+1) rest in replrec 0 (* [liftn_tomatch_stack]: a term to match has just been substituted by some constructor t = (ci x1...xn) and the terms x1 ... xn have been added to match; all pushed terms to match must be lifted by n (knowing that [Abstract] introduces a binder in the list of pushed terms to match). *) let rec liftn_tomatch_stack n depth = function | [] -> [] | Pushed ((c,tm),l,na)::rest -> let c = liftn n depth c in let tm = liftn_tomatch_type n depth tm in let l = List.map (fun i -> if i Alias (na,liftn n depth c,map_pair (liftn n depth) d) ::(liftn_tomatch_stack n depth rest) | NonDepAlias :: rest -> NonDepAlias :: liftn_tomatch_stack n depth rest | Abstract (i,d)::rest -> let i = if i x | x => x end] should be compiled into [match y with O => y | (S n) => match n with O => y | (S x) => x end end] and [match y with (S (S n)) => n | n => n end] into [match y with O => y | (S n0) => match n0 with O => y | (S n) => n end end] i.e. user names should be preserved and created names should not interfere with user names The exact names here are not important for typing (because they are put in pb.env and not in the rhs.rhs_env of branches. However, whether a name is Anonymous or not may have an effect on whether a generalization is done or not. *) let merge_name get_name obj = function | Anonymous -> get_name obj | na -> na let merge_names get_name = List.map2 (merge_name get_name) let get_names env sign eqns = let names1 = list_make (List.length sign) Anonymous in (* If any, we prefer names used in pats, from top to bottom *) let names2,aliasname = List.fold_right (fun (pats,pat_alias,eqn) (names,aliasname) -> (merge_names alias_of_pat pats names, merge_name (fun x -> x) pat_alias aliasname)) eqns (names1,Anonymous) in (* Otherwise, we take names from the parameters of the constructor but avoiding conflicts with user ids *) let allvars = List.fold_left (fun l (_,_,eqn) -> list_union l eqn.rhs.avoid_ids) [] eqns in let names3,_ = List.fold_left2 (fun (l,avoid) d na -> let na = merge_name (fun (na,_,t) -> Name (next_name_away (named_hd env t na) avoid)) d na in (na::l,(out_name na)::avoid)) ([],allvars) (List.rev sign) names2 in names3,aliasname (*****************************************************************) (* Recovering names for variables pushed to the rhs' environment *) (* We just factorized a match over a matrix of equations *) (* "C xi1 .. xin as xi" as a single match over "C y1 .. yn as y" *) (* We now replace the names y1 .. yn y by the actual names *) (* xi1 .. xin xi to be found in the i-th clause of the matrix *) let set_declaration_name x (_,c,t) = (x,c,t) let recover_initial_subpattern_names = List.map2 set_declaration_name let recover_alias_names get_name = List.map2 (fun x (_,c,t) ->(get_name x,c,t)) let push_rels_eqn sign eqn = {eqn with rhs = {eqn.rhs with rhs_env = push_rels sign eqn.rhs.rhs_env} } let push_rels_eqn_with_names sign eqn = let subpats = List.rev (list_firstn (List.length sign) eqn.patterns) in let subpatnames = List.map alias_of_pat subpats in let sign = recover_initial_subpattern_names subpatnames sign in push_rels_eqn sign eqn let push_generalized_decl_eqn env n (na,c,t) eqn = let na = match na with | Anonymous -> Anonymous | Name id -> pi1 (Environ.lookup_rel n eqn.rhs.rhs_env) in push_rels_eqn [(na,c,t)] eqn let drop_alias_eqn eqn = { eqn with alias_stack = List.tl eqn.alias_stack } let push_alias_eqn alias eqn = let aliasname = List.hd eqn.alias_stack in let eqn = drop_alias_eqn eqn in let alias = set_declaration_name aliasname alias in push_rels_eqn [alias] eqn (**********************************************************************) (* Functions to deal with elimination predicate *) (* Infering the predicate *) (* The problem to solve is the following: We match Gamma |- t : I(u01..u0q) against the following constructors: Gamma, x11...x1p1 |- C1(x11..x1p1) : I(u11..u1q) ... Gamma, xn1...xnpn |- Cn(xn1..xnp1) : I(un1..unq) Assume the types in the branches are the following Gamma, x11...x1p1 |- branch1 : T1 ... Gamma, xn1...xnpn |- branchn : Tn Assume the type of the global case expression is Gamma |- T The predicate has the form phi = [y1..yq][z:I(y1..yq)]psi and it has to satisfy the following n+1 equations: Gamma, x11...x1p1 |- (phi u11..u1q (C1 x11..x1p1)) = T1 ... Gamma, xn1...xnpn |- (phi un1..unq (Cn xn1..xnpn)) = Tn Gamma |- (phi u01..u0q t) = T Some hints: - Clearly, if xij occurs in Ti, then, a "match z with (Ci xi1..xipi) => ... end" or a "psi(yk)", with psi extracting xij from uik, should be inserted somewhere in Ti. - If T is undefined, an easy solution is to insert a "match z with (Ci xi1..xipi) => ... end" in front of each Ti - Otherwise, T1..Tn and T must be step by step unified, if some of them diverge, then try to replace the diverging subterm by one of y1..yq or z. - The main problem is what to do when an existential variables is encountered *) (* Propagation of user-provided predicate through compilation steps *) let rec map_predicate f k ccl = function | [] -> f k ccl | Pushed ((_,tm),_,na) :: rest -> let k' = length_of_tomatch_type_sign na tm in map_predicate f (k+k') ccl rest | (Alias _ | NonDepAlias) :: rest -> map_predicate f k ccl rest | Abstract _ :: rest -> map_predicate f (k+1) ccl rest let noccur_predicate_between n = map_predicate (noccur_between n) let liftn_predicate n = map_predicate (liftn n) let lift_predicate n = liftn_predicate n 1 let regeneralize_index_predicate n = map_predicate (relocate_index n 1) 0 let substnl_predicate sigma = map_predicate (substnl sigma) (* This is parallel bindings *) let subst_predicate (args,copt) ccl tms = let sigma = match copt with | None -> List.rev args | Some c -> c::(List.rev args) in substnl_predicate sigma 0 ccl tms let specialize_predicate_var (cur,typ,dep) tms ccl = let c = if dep<>Anonymous then Some cur else None in let l = match typ with | IsInd (_,IndType(_,realargs),names) -> if names<>[] then realargs else [] | NotInd _ -> [] in subst_predicate (l,c) ccl tms (*****************************************************************************) (* We have pred = [X:=realargs;x:=c]P typed in Gamma1, x:I(realargs), Gamma2 *) (* and we want to abstract P over y:t(x) typed in the same context to get *) (* *) (* pred' = [X:=realargs;x':=c](y':t(x'))P[y:=y'] *) (* *) (* We first need to lift t(x) s.t. it is typed in Gamma, X:=rargs, x' *) (* then we have to replace x by x' in t(x) and y by y' in P *) (*****************************************************************************) let generalize_predicate (names,na) ny d tms ccl = if na=Anonymous then anomaly "Undetected dependency"; let p = List.length names + 1 in let ccl = lift_predicate 1 ccl tms in regeneralize_index_predicate (ny+p+1) ccl tms (*****************************************************************************) (* We just matched over cur:ind(realargs) in the following matching problem *) (* *) (* env |- match cur tms return ccl with ... end *) (* *) (* and we want to build the predicate corresponding to the individual *) (* matching over cur *) (* *) (* pred = fun X:realargstyps x:ind(X)] PI tms.ccl *) (* *) (* where pred is computed by abstract_predicate and PI tms.ccl by *) (* extract_predicate *) (*****************************************************************************) let rec extract_predicate ccl = function | (Alias _ | NonDepAlias)::tms -> (* substitution already done in build_branch *) extract_predicate ccl tms | Abstract (i,d)::tms -> mkProd_wo_LetIn d (extract_predicate ccl tms) | Pushed ((cur,NotInd _),_,na)::tms -> let tms = if na<>Anonymous then lift_tomatch_stack 1 tms else tms in let pred = extract_predicate ccl tms in if na<>Anonymous then subst1 cur pred else pred | Pushed ((cur,IsInd (_,IndType(_,realargs),_)),_,na)::tms -> let realargs = List.rev realargs in let k = if na<>Anonymous then 1 else 0 in let tms = lift_tomatch_stack (List.length realargs + k) tms in let pred = extract_predicate ccl tms in substl (if na<>Anonymous then cur::realargs else realargs) pred | [] -> ccl let abstract_predicate env sigma indf cur realargs (names,na) tms ccl = let sign = make_arity_signature env true indf in (* n is the number of real args + 1 (+ possible let-ins in sign) *) let n = List.length sign in (* Before abstracting we generalize over cur and on those realargs *) (* that are rels, consistently with the specialization made in *) (* build_branch *) let tms = List.fold_right2 (fun par arg tomatch -> match kind_of_term par with | Rel i -> relocate_index_tomatch (i+n) (destRel arg) tomatch | _ -> tomatch) (realargs@[cur]) (extended_rel_list 0 sign) (lift_tomatch_stack n tms) in (* Pred is already dependent in the current term to match (if *) (* (na<>Anonymous) and its realargs; we just need to adjust it to *) (* full sign if dep in cur is not taken into account *) let ccl = if na <> Anonymous then ccl else lift_predicate 1 ccl tms in let pred = extract_predicate ccl tms in (* Build the predicate properly speaking *) let sign = List.map2 set_declaration_name (na::names) sign in it_mkLambda_or_LetIn_name env pred sign (* [expand_arg] is used by [specialize_predicate] if Yk denotes [Xk;xk] or [Xk], it replaces gamma, x1...xn, x1...xk Yk+1...Yn |- pred by gamma, x1...xn, x1...xk-1 [Xk;xk] Yk+1...Yn |- pred (if dep) or by gamma, x1...xn, x1...xk-1 [Xk] Yk+1...Yn |- pred (if not dep) *) let expand_arg tms (p,ccl) ((_,t),_,na) = let k = length_of_tomatch_type_sign na t in (p+k,liftn_predicate (k-1) (p+1) ccl tms) let adjust_impossible_cases pb pred tomatch submat = if submat = [] then match kind_of_term (whd_evar !(pb.evdref) pred) with | Evar (evk,_) when snd (evar_source evk !(pb.evdref)) = ImpossibleCase -> let default = (coq_unit_judge ()).uj_type in pb.evdref := Evd.define evk default !(pb.evdref); (* we add an "assert false" case *) let pats = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) tomatch in let aliasnames = map_succeed (function Alias _ | NonDepAlias -> Anonymous | _ -> failwith"") tomatch in [ { patterns = pats; rhs = { rhs_env = pb.env; rhs_vars = []; avoid_ids = []; it = None }; alias_stack = Anonymous::aliasnames; eqn_loc = dummy_loc; used = ref false } ] | _ -> submat else submat (*****************************************************************************) (* Let pred = PI [X;x:I(X)]. PI tms. P be a typing predicate for the *) (* following pattern-matching problem: *) (* *) (* Gamma |- match Pushed(c:I(V)) as x in I(X), tms return pred with...end *) (* *) (* where the branch with constructor Ci:(x1:T1)...(xn:Tn)->I(realargsi) *) (* is considered. Assume each Ti is some Ii(argsi) with Ti:PI Ui. sort_i *) (* We let subst = X:=realargsi;x:=Ci(x1,...,xn) and replace pred by *) (* *) (* pred' = PI [X1:Ui;x1:I1(X1)]...[Xn:Un;xn:In(Xn)]. (PI tms. P)[subst] *) (* *) (* s.t. the following well-typed sub-pattern-matching problem is obtained *) (* *) (* Gamma,x'1..x'n |- *) (* match *) (* Pushed(x'1) as x1 in I(X1), *) (* .., *) (* Pushed(x'n) as xn in I(Xn), *) (* tms *) (* return pred' *) (* with .. end *) (* *) (*****************************************************************************) let specialize_predicate newtomatchs (names,depna) arsign cs tms ccl = (* Assume some gamma st: gamma |- PI [X,x:I(X)]. PI tms. ccl *) let nrealargs = List.length names in let k = nrealargs + (if depna<>Anonymous then 1 else 0) in (* We adjust pred st: gamma, x1..xn |- PI [X,x:I(X)]. PI tms. ccl' *) (* so that x can later be instantiated by Ci(x1..xn) *) (* and X by the realargs for Ci *) let n = cs.cs_nargs in let ccl' = liftn_predicate n (k+1) ccl tms in (* We prepare the substitution of X and x:I(X) *) let realargsi = if nrealargs <> 0 then adjust_subst_to_rel_context arsign (Array.to_list cs.cs_concl_realargs) else [] in let copti = if depna<>Anonymous then Some (build_dependent_constructor cs) else None in (* The substituends realargsi, copti are all defined in gamma, x1...xn *) (* We need _parallel_ bindings to get gamma, x1...xn |- PI tms. ccl'' *) (* Note: applying the substitution in tms is not important (is it sure?) *) let ccl'' = whd_betaiota Evd.empty (subst_predicate (realargsi, copti) ccl' tms) in (* We adjust ccl st: gamma, x'1..x'n, x1..xn, tms |- ccl'' *) let ccl''' = liftn_predicate n (n+1) ccl'' tms in (* We finally get gamma,x'1..x'n,x |- [X1;x1:I(X1)]..[Xn;xn:I(Xn)]pred'''*) snd (List.fold_left (expand_arg tms) (1,ccl''') newtomatchs) let find_predicate loc env evdref p current (IndType (indf,realargs)) dep tms = let pred = abstract_predicate env !evdref indf current realargs dep tms p in (pred, whd_betaiota !evdref (applist (pred, realargs@[current]))) (* Take into account that a type has been discovered to be inductive, leading to more dependencies in the predicate if the type has indices *) let adjust_predicate_from_tomatch tomatch (current,typ as ct) pb = let ((_,oldtyp),deps,na) = tomatch in match typ, oldtyp with | IsInd (_,_,names), NotInd _ -> let k = if na <> Anonymous then 2 else 1 in let n = List.length names in { pb with pred = liftn_predicate n k pb.pred pb.tomatch }, (ct,List.map (fun i -> if i >= k then i+n else i) deps,na) | _ -> pb, (ct,deps,na) (* Remove commutative cuts that turn out to be non-dependent after some evars have been instantiated *) let rec ungeneralize n ng body = match kind_of_term body with | Lambda (_,_,c) when ng = 0 -> subst1 (mkRel n) c | Lambda (na,t,c) -> (* We traverse an inner generalization *) mkLambda (na,t,ungeneralize (n+1) (ng-1) c) | LetIn (na,b,t,c) -> (* We traverse an alias *) mkLetIn (na,b,t,ungeneralize (n+1) ng c) | Case (ci,p,c,brs) -> (* We traverse a split *) let p = let sign,p = decompose_lam_assum p in let sign2,p = decompose_prod_n_assum ng p in let p = prod_applist p [mkRel (n+List.length sign+ng)] in it_mkLambda_or_LetIn (it_mkProd_or_LetIn p sign2) sign in mkCase (ci,p,c,array_map2 (fun q c -> let sign,b = decompose_lam_n_assum q c in it_mkLambda_or_LetIn (ungeneralize (n+q) ng b) sign) ci.ci_cstr_ndecls brs) | App (f,args) -> (* We traverse an inner generalization *) assert (isCase f); mkApp (ungeneralize n (ng+Array.length args) f,args) | _ -> assert false let ungeneralize_branch n k (sign,body) cs = (sign,ungeneralize (n+cs.cs_nargs) k body) let postprocess_dependencies evd tocheck brs tomatch pred deps cs = let rec aux k brs tomatch pred tocheck deps = match deps, tomatch with | [], _ -> brs,tomatch,pred,[] | n::deps, Abstract (i,d) :: tomatch -> let d = map_rel_declaration (nf_evar evd) d in if List.exists (fun c -> dependent_decl (lift k c) d) tocheck || pi2 d <> None then (* Dependency in the current term to match and its dependencies is real *) let brs,tomatch,pred,inst = aux (k+1) brs tomatch pred (mkRel n::tocheck) deps in let inst = if pi2 d = None then mkRel n::inst else inst in brs, Abstract (i,d) :: tomatch, pred, inst else (* Finally, no dependency remains, so, we can replace the generalized *) (* terms by its actual value in both the remaining terms to match and *) (* the bodies of the Case *) let pred = lift_predicate (-1) pred tomatch in let tomatch = relocate_index_tomatch 1 (n+1) tomatch in let tomatch = lift_tomatch_stack (-1) tomatch in let brs = array_map2 (ungeneralize_branch n k) brs cs in aux k brs tomatch pred tocheck deps | _ -> assert false in aux 0 brs tomatch pred tocheck deps (************************************************************************) (* Sorting equations by constructor *) let rec irrefutable env = function | PatVar (_,name) -> true | PatCstr (_,cstr,args,_) -> let ind = inductive_of_constructor cstr in let (_,mip) = Inductive.lookup_mind_specif env ind in let one_constr = Array.length mip.mind_user_lc = 1 in one_constr & List.for_all (irrefutable env) args let first_clause_irrefutable env = function | eqn::mat -> List.for_all (irrefutable env) eqn.patterns | _ -> false let group_equations pb ind current cstrs mat = let mat = if first_clause_irrefutable pb.env mat then [List.hd mat] else mat in let brs = Array.create (Array.length cstrs) [] in let only_default = ref true in let _ = List.fold_right (* To be sure it's from bottom to top *) (fun eqn () -> let rest = remove_current_pattern eqn in let pat = current_pattern eqn in match check_and_adjust_constructor pb.env ind cstrs pat with | PatVar (_,name) -> (* This is a default clause that we expand *) for i=1 to Array.length cstrs do let args = make_anonymous_patvars cstrs.(i-1).cs_nargs in brs.(i-1) <- (args, name, rest) :: brs.(i-1) done | PatCstr (loc,((_,i)),args,name) -> (* This is a regular clause *) only_default := false; brs.(i-1) <- (args, name, rest) :: brs.(i-1)) mat () in (brs,!only_default) (************************************************************************) (* Here starts the pattern-matching compilation algorithm *) (* Abstracting over dependent subterms to match *) let rec generalize_problem names pb = function | [] -> pb, [] | i::l -> let (na,b,t as d) = map_rel_declaration (lift i) (Environ.lookup_rel i pb.env) in let pb',deps = generalize_problem names pb l in if na = Anonymous & b <> None then pb',deps else let d = on_pi3 (whd_betaiota !(pb.evdref)) d in (* for better rendering *) let tomatch = lift_tomatch_stack 1 pb'.tomatch in let tomatch = relocate_index_tomatch (i+1) 1 tomatch in { pb' with tomatch = Abstract (i,d) :: tomatch; pred = generalize_predicate names i d pb'.tomatch pb'.pred }, i::deps (* No more patterns: typing the right-hand side of equations *) let build_leaf pb = let rhs = extract_rhs pb in let j = pb.typing_function (mk_tycon pb.pred) rhs.rhs_env pb.evdref rhs.it in j_nf_evar !(pb.evdref) j (* Build the sub-pattern-matching problem for a given branch "C x1..xn as x" *) let build_branch current realargs deps (realnames,curname) pb arsign eqns const_info = (* We remember that we descend through constructor C *) let history = push_history_pattern const_info.cs_nargs const_info.cs_cstr pb.history in (* We prepare the matching on x1:T1 .. xn:Tn using some heuristic to *) (* build the name x1..xn from the names present in the equations *) (* that had matched constructor C *) let cs_args = const_info.cs_args in let names,aliasname = get_names pb.env cs_args eqns in let typs = List.map2 (fun (_,c,t) na -> (na,c,t)) cs_args names in (* We build the matrix obtained by expanding the matching on *) (* "C x1..xn as x" followed by a residual matching on eqn into *) (* a matching on "x1 .. xn eqn" *) let submat = List.map (fun (tms,_,eqn) -> prepend_pattern tms eqn) eqns in (* We adjust the terms to match in the context they will be once the *) (* context [x1:T1,..,xn:Tn] will have been pushed on the current env *) let typs' = list_map_i (fun i d -> (mkRel i,map_rel_declaration (lift i) d)) 1 typs in let extenv = push_rels typs pb.env in let typs' = List.map (fun (c,d) -> (c,extract_inductive_data extenv !(pb.evdref) d,d)) typs' in (* We compute over which of x(i+1)..xn and x matching on xi will need a *) (* generalization *) let dep_sign = find_dependencies_signature (dependencies_in_rhs const_info.cs_nargs current pb.tomatch eqns) (List.rev typs') in (* The dependent term to subst in the types of the remaining UnPushed terms is relative to the current context enriched by topushs *) let ci = build_dependent_constructor const_info in (* Current context Gamma has the form Gamma1;cur:I(realargs);Gamma2 *) (* We go from Gamma |- PI tms. pred to *) (* Gamma;x1..xn;curalias:I(x1..xn) |- PI tms'. pred' *) (* where, in tms and pred, those realargs that are vars are *) (* replaced by the corresponding xi and cur replaced by curalias *) let cirealargs = Array.to_list const_info.cs_concl_realargs in (* Do the specialization for terms to match *) let tomatch = List.fold_right2 (fun par arg tomatch -> match kind_of_term par with | Rel i -> replace_tomatch (i+const_info.cs_nargs) arg tomatch | _ -> tomatch) (current::realargs) (ci::cirealargs) (lift_tomatch_stack const_info.cs_nargs pb.tomatch) in let pred_is_not_dep = noccur_predicate_between 1 (List.length realnames + 1) pb.pred tomatch in let typs' = List.map2 (fun (tm,(tmtyp,_),(na,_,_)) deps -> let na = match curname with | Name _ -> (if na <> Anonymous then na else curname) | Anonymous -> if deps = [] && pred_is_not_dep then Anonymous else force_name na in ((tm,tmtyp),deps,na)) typs' (List.rev dep_sign) in (* Do the specialization for the predicate *) let pred = specialize_predicate typs' (realnames,curname) arsign const_info tomatch pb.pred in let currents = List.map (fun x -> Pushed x) typs' in let alias = if aliasname = Anonymous then NonDepAlias else let cur_alias = lift const_info.cs_nargs current in let ind = appvect ( applist (mkInd (inductive_of_constructor const_info.cs_cstr), List.map (lift const_info.cs_nargs) const_info.cs_params), const_info.cs_concl_realargs) in Alias (aliasname,cur_alias,(ci,ind)) in let tomatch = List.rev_append (alias :: currents) tomatch in let submat = adjust_impossible_cases pb pred tomatch submat in if submat = [] then raise_pattern_matching_error (dummy_loc, pb.env, NonExhaustive (complete_history history)); typs, { pb with env = extenv; tomatch = tomatch; pred = pred; history = history; mat = List.map (push_rels_eqn_with_names typs) submat } (********************************************************************** INVARIANT: pb = { env, pred, tomatch, mat, ...} tomatch = list of Pushed (c:T), Abstract (na:T), Alias (c:T) or NonDepAlias all terms and types in Pushed, Abstract and Alias are relative to env enriched by the Abstract coming before *) (**********************************************************************) (* Main compiling descent *) let rec compile pb = match pb.tomatch with | Pushed cur :: rest -> match_current { pb with tomatch = rest } cur | Alias x :: rest -> compile_alias pb x rest | NonDepAlias :: rest -> compile_non_dep_alias pb rest | Abstract (i,d) :: rest -> compile_generalization pb i d rest | [] -> build_leaf pb (* Case splitting *) and match_current pb tomatch = let tm = adjust_tomatch_to_pattern pb tomatch in let pb,tomatch = adjust_predicate_from_tomatch tomatch tm pb in let ((current,typ),deps,dep) = tomatch in match typ with | NotInd (_,typ) -> check_all_variables typ pb.mat; shift_problem tomatch pb | IsInd (_,(IndType(indf,realargs) as indt),names) -> let mind,_ = dest_ind_family indf in let cstrs = get_constructors pb.env indf in let arsign, _ = get_arity pb.env indf in let eqns,onlydflt = group_equations pb mind current cstrs pb.mat in if (Array.length cstrs <> 0 or pb.mat <> []) & onlydflt then shift_problem tomatch pb else (* We generalize over terms depending on current term to match *) let pb,deps = generalize_problem (names,dep) pb deps in (* We compile branches *) let brvals = array_map2 (compile_branch current realargs (names,dep) deps pb arsign) eqns cstrs in (* We build the (elementary) case analysis *) let depstocheck = current::binding_vars_of_inductive typ in let brvals,tomatch,pred,inst = postprocess_dependencies !(pb.evdref) depstocheck brvals pb.tomatch pb.pred deps cstrs in let brvals = Array.map (fun (sign,body) -> it_mkLambda_or_LetIn body sign) brvals in let (pred,typ) = find_predicate pb.caseloc pb.env pb.evdref pred current indt (names,dep) tomatch in let ci = make_case_info pb.env mind pb.casestyle in let pred = nf_betaiota !(pb.evdref) pred in let case = mkCase (ci,pred,current,brvals) in Typing.check_allowed_sort pb.env !(pb.evdref) mind current pred; { uj_val = applist (case, inst); uj_type = prod_applist typ inst } (* Building the sub-problem when all patterns are variables *) and shift_problem ((current,t),_,na) pb = let ty = type_of_tomatch t in let tomatch = lift_tomatch_stack 1 pb.tomatch in let pred = specialize_predicate_var (current,t,na) pb.tomatch pb.pred in let pb = { pb with env = push_rel (na,Some current,ty) pb.env; tomatch = tomatch; pred = lift_predicate 1 pred tomatch; history = pop_history pb.history; mat = List.map (push_current_pattern (current,ty)) pb.mat } in let j = compile pb in { uj_val = subst1 current j.uj_val; uj_type = subst1 current j.uj_type } (* Building the sub-problem when all patterns are variables *) and compile_branch current realargs names deps pb arsign eqns cstr = let sign, pb = build_branch current realargs deps names pb arsign eqns cstr in sign, (compile pb).uj_val (* Abstract over a declaration before continuing splitting *) and compile_generalization pb i d rest = let pb = { pb with env = push_rel d pb.env; tomatch = rest; mat = List.map (push_generalized_decl_eqn pb.env i d) pb.mat } in let j = compile pb in { uj_val = mkLambda_or_LetIn d j.uj_val; uj_type = mkProd_wo_LetIn d j.uj_type } and compile_alias pb (na,orig,(expanded,expanded_typ)) rest = let f c t = let alias = (na,Some c,t) in let pb = { pb with env = push_rel alias pb.env; tomatch = lift_tomatch_stack 1 rest; pred = lift_predicate 1 pb.pred pb.tomatch; history = pop_history_pattern pb.history; mat = List.map (push_alias_eqn alias) pb.mat } in let j = compile pb in { uj_val = if isRel c || isVar c || count_occurrences (mkRel 1) j.uj_val <= 1 then subst1 c j.uj_val else mkLetIn (na,c,t,j.uj_val); uj_type = subst1 c j.uj_type } in if isRel orig or isVar orig then (* Try to compile first using non expanded alias *) try f orig (Retyping.get_type_of pb.env !(pb.evdref) orig) with e when precatchable_exception e -> (* Try then to compile using expanded alias *) f expanded expanded_typ else (* Try to compile first using expanded alias *) try f expanded expanded_typ with e when precatchable_exception e -> (* Try then to compile using non expanded alias *) f orig (Retyping.get_type_of pb.env !(pb.evdref) orig) (* Remember that a non-trivial pattern has been consumed *) and compile_non_dep_alias pb rest = let pb = { pb with tomatch = rest; history = pop_history_pattern pb.history; mat = List.map drop_alias_eqn pb.mat } in compile pb (* pour les alias des initiaux, enrichir les env de ce qu'il faut et substituer après par les initiaux *) (**************************************************************************) (* Preparation of the pattern-matching problem *) (* builds the matrix of equations testing that each eqn has n patterns * and linearizing the _ patterns. * Syntactic correctness has already been done in astterm *) let matx_of_eqns env tomatchl eqns = let build_eqn (loc,ids,lpat,rhs) = let initial_lpat,initial_rhs = lpat,rhs in let initial_rhs = rhs in let rhs = { rhs_env = env; rhs_vars = free_glob_vars initial_rhs; avoid_ids = ids@(ids_of_named_context (named_context env)); it = Some initial_rhs } in { patterns = initial_lpat; alias_stack = []; eqn_loc = loc; used = ref false; rhs = rhs } in List.map build_eqn eqns (***************** Building an inversion predicate ************************) (* Let "match t1 in I1 u11..u1n_1 ... tm in Im um1..umn_m with ... end : T" be a pattern-matching problem. We assume that each uij can be decomposed under the form pij(vij1..vijq_ij) where pij(aij1..aijq_ij) is a pattern depending on some variables aijk and the vijk are instances of these variables. We also assume that each ti has the form of a pattern qi(wi1..wiq_i) where qi(bi1..biq_i) is a pattern depending on some variables bik and the wik are instances of these variables (in practice, there is no reason that ti is already constructed and the qi will be degenerated). We then look for a type U(..a1jk..b1 .. ..amjk..bm) so that T = U(..v1jk..t1 .. ..vmjk..tm). This a higher-order matching problem with a priori different solutions (one of them if T itself!). We finally invert the uij and the ti and build the return clause phi(x11..x1n_1y1..xm1..xmn_mym) = match x11..x1n_1 y1 .. xm1..xmn_m ym with | p11..p1n_1 q1 .. pm1..pmn_m qm => U(..a1jk..b1 .. ..amjk..bm) | _ .. _ _ .. _ .. _ _ => True end so that "phi(u11..u1n_1t1..um1..umn_mtm) = T" (note that the clause returning True never happens and any inhabited type can be put instead). *) let adjust_to_extended_env_and_remove_deps env extenv subst t = let n = rel_context_length (rel_context env) in let n' = rel_context_length (rel_context extenv) in (* We first remove the bindings that are dependently typed (they are difficult to manage and it is not sure these are so useful in practice); Notes: - [subst] is made of pairs [(id,u)] where id is a name in [extenv] and [u] a term typed in [env]; - [subst0] is made of items [(p,u,(u,ty))] where [ty] is the type of [u] and both are adjusted to [extenv] while [p] is the index of [id] in [extenv] (after expansion of the aliases) *) let subst0 = map_succeed (fun (x,u) -> (* d1 ... dn dn+1 ... dn'-p+1 ... dn' *) (* \--env-/ (= x:ty) *) (* \--------------extenv------------/ *) let (p,_,_) = lookup_rel_id x (rel_context extenv) in let rec traverse_local_defs p = match pi2 (lookup_rel p extenv) with | Some c -> assert (isRel c); traverse_local_defs (p + destRel c) | None -> p in let p = traverse_local_defs p in let u = lift (n'-n) u in (p,u,expand_vars_in_term extenv u)) subst in let t0 = lift (n'-n) t in (subst0,t0) let push_binder d (k,env,subst) = (k+1,push_rel d env,List.map (fun (na,u,d) -> (na,lift 1 u,d)) subst) let rec list_assoc_in_triple x = function [] -> raise Not_found | (a,b,_)::l -> if compare a x = 0 then b else list_assoc_in_triple x l (* Let vijk and ti be a set of dependent terms and T a type, all * defined in some environment env. The vijk and ti are supposed to be * instances for variables aijk and bi. * * [abstract_tycon Gamma0 Sigma subst T Gamma] looks for U(..v1jk..t1 .. ..vmjk..tm) * defined in some extended context * "Gamma0, ..a1jk:V1jk.. b1:W1 .. ..amjk:Vmjk.. bm:Wm" * such that env |- T = U(..v1jk..t1 .. ..vmjk..tm). To not commit to * a particular solution, we replace each subterm t in T that unifies with * a subset u1..ul of the vijk and ti by a special evar * ?id(x=t;c1:=c1,..,cl=cl) defined in context Gamma0,x,c1,...,cl |- ?id * (where the c1..cl are the aijk and bi matching the u1..ul), and * similarly for each ti. *) let abstract_tycon loc env evdref subst _tycon extenv t = let sigma = !evdref in let t = nf_betaiota sigma t in (* it helps in some cases to remove K-redex *) let subst0,t0 = adjust_to_extended_env_and_remove_deps env extenv subst t in (* We traverse the type T of the original problem Xi looking for subterms that match the non-constructor part of the constraints (this part is in subst); these subterms are the "good" subterms and we replace them by an evar that may depend (and only depend) on the corresponding convertible subterms of the substitution *) let rec aux (k,env,subst as x) t = let t = whd_evar !evdref t in match kind_of_term t with | Rel n when pi2 (lookup_rel n env) <> None -> map_constr_with_full_binders push_binder aux x t | Evar ev -> let ty = get_type_of env sigma t in let inst = list_map_i (fun i _ -> try list_assoc_in_triple i subst0 with Not_found -> mkRel i) 1 (rel_context env) in let ev = e_new_evar evdref env ~src:(loc, CasesType) ty in evdref := add_conv_pb (Reduction.CONV,env,substl inst ev,t) !evdref; ev | _ -> let good = List.filter (fun (_,u,_) -> is_conv_leq env sigma t u) subst in if good <> [] then let u = pi3 (List.hd good) in (* u is in extenv *) let vl = List.map pi1 good in let ty = lift (-k) (aux x (get_type_of env !evdref t)) in let depvl = free_rels ty in let inst = list_map_i (fun i _ -> if List.mem i vl then u else mkRel i) 1 (rel_context extenv) in let rel_filter = List.map (fun a -> not (isRel a) || dependent a u || Intset.mem (destRel a) depvl) inst in let named_filter = List.map (fun (id,_,_) -> dependent (mkVar id) u) (named_context extenv) in let filter = rel_filter@named_filter in let candidates = u :: List.map mkRel vl in let ev = e_new_evar evdref extenv ~src:(loc, CasesType) ~filter ~candidates ty in lift k ev else map_constr_with_full_binders push_binder aux x t in aux (0,extenv,subst0) t0 let build_tycon loc env tycon_env subst tycon extenv evdref t = let t,tt = match t with | None -> (* This is the situation we are building a return predicate and we are in an impossible branch *) let n = rel_context_length (rel_context env) in let n' = rel_context_length (rel_context tycon_env) in let tt = new_Type () in let impossible_case_type = e_new_evar evdref env ~src:(loc,ImpossibleCase) tt in (lift (n'-n) impossible_case_type, tt) | Some t -> let t = abstract_tycon loc tycon_env evdref subst tycon extenv t in let evd,tt = Typing.e_type_of extenv !evdref t in evdref := evd; (t,tt) in { uj_val = t; uj_type = tt } (* For a multiple pattern-matching problem Xi on t1..tn with return * type T, [build_inversion_problem Gamma Sigma (t1..tn) T] builds a return * predicate for Xi that is itself made by an auxiliary * pattern-matching problem of which the first clause reveals the * pattern structure of the constraints on the inductive types of the t1..tn, * and the second clause is a wildcard clause for catching the * impossible cases. See above "Building an inversion predicate" for * further explanations *) let build_inversion_problem loc env sigma tms t = let make_patvar t (subst,avoid) = let id = next_name_away (named_hd env t Anonymous) avoid in PatVar (dummy_loc,Name id), ((id,t)::subst, id::avoid) in let rec reveal_pattern t (subst,avoid as acc) = match kind_of_term (whd_betadeltaiota env sigma t) with | Construct cstr -> PatCstr (dummy_loc,cstr,[],Anonymous), acc | App (f,v) when isConstruct f -> let cstr = destConstruct f in let n = constructor_nrealargs env cstr in let l = list_lastn n (Array.to_list v) in let l,acc = list_fold_map' reveal_pattern l acc in PatCstr (dummy_loc,cstr,l,Anonymous), acc | _ -> make_patvar t acc in let rec aux n env acc_sign tms acc = match tms with | [] -> [], acc_sign, acc | (t, IsInd (_,IndType(indf,realargs),_)) :: tms -> let patl,acc = list_fold_map' reveal_pattern realargs acc in let pat,acc = make_patvar t acc in let indf' = lift_inductive_family n indf in let sign = make_arity_signature env true indf' in let sign = recover_alias_names alias_of_pat (pat :: List.rev patl) sign in let p = List.length realargs in let env' = push_rels sign env in let patl',acc_sign,acc = aux (n+p+1) env' (sign@acc_sign) tms acc in patl@pat::patl',acc_sign,acc | (t, NotInd (bo,typ)) :: tms -> let pat,acc = make_patvar t acc in let d = (alias_of_pat pat,None,t) in let patl,acc_sign,acc = aux (n+1) (push_rel d env) (d::acc_sign) tms acc in pat::patl,acc_sign,acc in let avoid0 = ids_of_context env in (* [patl] is a list of patterns revealing the substructure of constructors present in the constraints on the type of the multiple terms t1..tn that are matched in the original problem; [subst] is the substitution of the free pattern variables in [patl] that returns the non-constructor parts of the constraints. Especially, if the ti has type I ui1..uin_i, and the patterns associated to ti are pi1..pin_i, then subst(pij) is uij; the substitution is useful to recognize which subterms of the whole type T of the original problem have to be abstracted *) let patl,sign,(subst,avoid) = aux 0 env [] tms ([],avoid0) in let n = List.length sign in let decls = list_map_i (fun i d -> (mkRel i,map_rel_declaration (lift i) d)) 1 sign in let pb_env = push_rels sign env in let decls = List.map (fun (c,d) -> (c,extract_inductive_data pb_env sigma d,d)) decls in let decls = List.rev decls in let dep_sign = find_dependencies_signature (list_make n true) decls in let sub_tms = List.map2 (fun deps (tm,(tmtyp,_),(na,b,t)) -> let na = if deps = [] then Anonymous else force_name na in Pushed ((tm,tmtyp),deps,na)) dep_sign decls in let subst = List.map (fun (na,t) -> (na,lift n t)) subst in (* [eqn1] is the first clause of the auxiliary pattern-matching that serves as skeleton for the return type: [patl] is the substructure of constructors extracted from the list of constraints on the inductive types of the multiple terms matched in the original pattern-matching problem Xi *) let eqn1 = { patterns = patl; alias_stack = []; eqn_loc = dummy_loc; used = ref false; rhs = { rhs_env = pb_env; (* we assume all vars are used; in practice we discard dependent vars so that the field rhs_vars is normally not used *) rhs_vars = List.map fst subst; avoid_ids = avoid; it = Some (lift n t) } } in (* [eqn2] is the default clause of the auxiliary pattern-matching: it will catch the clauses of the original pattern-matching problem Xi whose type constraints are incompatible with the constraints on the inductive types of the multiple terms matched in Xi *) let eqn2 = { patterns = List.map (fun _ -> PatVar (dummy_loc,Anonymous)) patl; alias_stack = []; eqn_loc = dummy_loc; used = ref false; rhs = { rhs_env = pb_env; rhs_vars = []; avoid_ids = avoid0; it = None } } in (* [pb] is the auxiliary pattern-matching serving as skeleton for the return type of the original problem Xi *) let evdref = ref sigma in let pb = { env = pb_env; evdref = evdref; pred = new_Type(); tomatch = sub_tms; history = start_history n; mat = [eqn1;eqn2]; caseloc = loc; casestyle = RegularStyle; typing_function = build_tycon loc env pb_env subst} in let pred = (compile pb).uj_val in (!evdref,pred) (* Here, [pred] is assumed to be in the context built from all *) (* realargs and terms to match *) let build_initial_predicate arsign pred = let rec buildrec n pred tmnames = function | [] -> List.rev tmnames,pred | ((na,c,t)::realdecls)::lnames -> let n' = n + List.length realdecls in buildrec (n'+1) pred (force_name na::tmnames) lnames | _ -> assert false in buildrec 0 pred [] (List.rev arsign) let extract_arity_signature env0 tomatchl tmsign = let get_one_sign n tm (na,t) = match tm with | NotInd (bo,typ) -> (match t with | None -> [na,Option.map (lift n) bo,lift n typ] | Some (loc,_,_,_) -> user_err_loc (loc,"", str"Unexpected type annotation for a term of non inductive type.")) | IsInd (term,IndType(indf,realargs),_) -> let indf' = lift_inductive_family n indf in let (ind,_) = dest_ind_family indf' in let nparams_ctxt,nrealargs_ctxt = inductive_nargs env0 ind in let arsign = fst (get_arity env0 indf') in let realnal = match t with | Some (loc,ind',nparams,realnal) -> if ind <> ind' then user_err_loc (loc,"",str "Wrong inductive type."); if nparams_ctxt <> nparams or nrealargs_ctxt <> List.length realnal then anomaly "Ill-formed 'in' clause in cases"; List.rev realnal | None -> list_make nrealargs_ctxt Anonymous in (na,None,build_dependent_inductive env0 indf') ::(List.map2 (fun x (_,c,t) ->(x,c,t)) realnal arsign) in let rec buildrec n = function | [],[] -> [] | (_,tm)::ltm, (_,x)::tmsign -> let l = get_one_sign n tm x in l :: buildrec (n + List.length l) (ltm,tmsign) | _ -> assert false in List.rev (buildrec 0 (tomatchl,tmsign)) let inh_conv_coerce_to_tycon loc env evdref j tycon = match tycon with | Some p -> let (evd',j) = Coercion.inh_conv_coerce_to loc env !evdref j p in evdref := evd'; j | None -> j (* We put the tycon inside the arity signature, possibly discovering dependencies. *) let prepare_predicate_from_arsign_tycon loc tomatchs arsign c = let nar = List.fold_left (fun n sign -> List.length sign + n) 0 arsign in let subst, len = List.fold_left2 (fun (subst, len) (tm, tmtype) sign -> let signlen = List.length sign in match kind_of_term tm with | Rel n when dependent tm c && signlen = 1 (* The term to match is not of a dependent type itself *) -> ((n, len) :: subst, len - signlen) | Rel n when signlen > 1 (* The term is of a dependent type, maybe some variable in its type appears in the tycon. *) -> (match tmtype with NotInd _ -> (subst, len - signlen) | IsInd (_, IndType(indf,realargs),_) -> let subst = if dependent tm c && List.for_all isRel realargs then (n, 1) :: subst else subst in List.fold_left (fun (subst, len) arg -> match kind_of_term arg with | Rel n when dependent arg c -> ((n, len) :: subst, pred len) | _ -> (subst, pred len)) (subst, len) realargs) | _ -> (subst, len - signlen)) ([], nar) tomatchs arsign in let rec predicate lift c = match kind_of_term c with | Rel n when n > lift -> (try (* Make the predicate dependent on the matched variable *) let idx = List.assoc (n - lift) subst in mkRel (idx + lift) with Not_found -> (* A variable that is not matched, lift over the arsign. *) mkRel (n + nar)) | _ -> map_constr_with_binders succ predicate lift c in predicate 0 c (* Builds the predicate. If the predicate is dependent, its context is * made of 1+nrealargs assumptions for each matched term in an inductive * type and 1 assumption for each term not _syntactically_ in an * inductive type. * Each matched terms are independently considered dependent or not. * A type constraint but no annotation case: we try to specialize the * tycon to make the predicate if it is not closed. *) let prepare_predicate loc typing_fun sigma env tomatchs arsign tycon pred = let preds = match pred, tycon with (* No type annotation *) | None, Some (None, t) when not (noccur_with_meta 0 max_int t) -> (* If the tycon is not closed w.r.t real variables, we try *) (* two different strategies *) (* First strategy: we abstract the tycon wrt to the dependencies *) let pred1 = prepare_predicate_from_arsign_tycon loc tomatchs arsign t in (* Second strategy: we build an "inversion" predicate *) let sigma2,pred2 = build_inversion_problem loc env sigma tomatchs t in [sigma, pred1; sigma2, pred2] | None, _ -> (* No dependent type constraint, or no constraints at all: *) (* we use two strategies *) let sigma,t = match tycon with | Some (None, t) -> sigma,t | _ -> new_type_evar sigma env ~src:(loc, CasesType) in (* First strategy: we build an "inversion" predicate *) let sigma1,pred1 = build_inversion_problem loc env sigma tomatchs t in (* Second strategy: we directly use the evar as a non dependent pred *) let pred2 = lift (List.length (List.flatten arsign)) t in [sigma1, pred1; sigma, pred2] (* Some type annotation *) | Some rtntyp, _ -> (* We extract the signature of the arity *) let envar = List.fold_right push_rels arsign env in let sigma, newt = new_sort_variable sigma in let evdref = ref sigma in let predcclj = typing_fun (mk_tycon (mkSort newt)) envar evdref rtntyp in let sigma = Option.cata (fun tycon -> let na = Name (id_of_string "x") in let tms = List.map (fun tm -> Pushed(tm,[],na)) tomatchs in let predinst = extract_predicate predcclj.uj_val tms in Coercion.inh_conv_coerces_to loc env !evdref predinst tycon) !evdref tycon in let predccl = (j_nf_evar sigma predcclj).uj_val in [sigma, predccl] in List.map (fun (sigma,pred) -> let (nal,pred) = build_initial_predicate arsign pred in sigma,nal,pred) preds (**************************************************************************) (* Main entry of the matching compilation *) let compile_cases loc style (typing_fun, evdref) tycon env (predopt, tomatchl, eqns) = (* We build the matrix of patterns and right-hand side *) let matx = matx_of_eqns env tomatchl eqns in (* We build the vector of terms to match consistently with the *) (* constructors found in patterns *) let tomatchs = coerce_to_indtype typing_fun evdref env matx tomatchl in (* If an elimination predicate is provided, we check it is compatible with the type of arguments to match; if none is provided, we build alternative possible predicates *) let arsign = extract_arity_signature env tomatchs tomatchl in let preds = prepare_predicate loc typing_fun !evdref env tomatchs arsign tycon predopt in let compile_for_one_predicate (sigma,nal,pred) = (* We push the initial terms to match and push their alias to rhs' envs *) (* names of aliases will be recovered from patterns (hence Anonymous *) (* here) *) let out_tmt na = function NotInd (c,t) -> (na,c,t) | IsInd (typ,_,_) -> (na,None,typ) in let typs = List.map2 (fun na (tm,tmt) -> (tm,out_tmt na tmt)) nal tomatchs in let typs = List.map (fun (c,d) -> (c,extract_inductive_data env sigma d,d)) typs in let dep_sign = find_dependencies_signature (list_make (List.length typs) true) typs in let typs' = list_map3 (fun (tm,tmt) deps na -> let deps = if not (isRel tm) then [] else deps in ((tm,tmt),deps,na)) tomatchs dep_sign nal in let initial_pushed = List.map (fun x -> Pushed x) typs' in (* A typing function that provides with a canonical term for absurd cases*) let typing_fun tycon env evdref = function | Some t -> typing_fun tycon env evdref t | None -> coq_unit_judge () in let myevdref = ref sigma in let pb = { env = env; evdref = myevdref; pred = pred; tomatch = initial_pushed; history = start_history (List.length initial_pushed); mat = matx; caseloc = loc; casestyle = style; typing_function = typing_fun } in let j = compile pb in evdref := !myevdref; j in (* Return the term compiled with the first possible elimination *) (* predicate for which the compilation succeeds *) let j = list_try_compile compile_for_one_predicate preds in (* We check for unused patterns *) List.iter (check_unused_pattern env) matx; (* We coerce to the tycon (if an elim predicate was provided) *) inh_conv_coerce_to_tycon loc env evdref j tycon end